# Thread: simple question on R module homomorphism

1. ## simple question on R module homomorphism

Let f: M ---> N be a R module homomorphism
P and Q are submodules of M and N respectively.
Show that:
(i) f(P) ={ f(p): p belongs to P} is a submodule of N
(ii) f^(-1)(Q) ={ m in M: f(m) belongs to Q} is a submodule of M
(iii) What are f(M) and f^(-1)(0)?

I have already proved part (i) and (ii)
My problem is part (iii). I don't know how to do it, it looks easy but I think it is a trick question. So, I have to bother you to help me on part(iii)

2. $f(M)$ is the image of f. $f^{-1}\{0\}$ is the kernal of f.
3. Yeah, I think you are just supposed to note that those are two very special cases of what you proved in i) and ii). If f(M)=N f is surjective (onto) and if $f^{-1}(0)=\{0\}$ then f is injective (one to one).